Optimal control of unsteady compressible viscous flows

The control of complex, unsteady flows is a pacing technology for advances in fluid mechanics. Recently, optimal control theory has become popular as a means of predicting best case controls that can guide the design of practical flow control systems. However, most of the prior work in this area has focused on incompressible flow which precludes many of the important physical flow phenomena that must be controlled in practice including the coupling of fluid dynamics, acoustics, and heat transfer. This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady two‐dimensional compressible Navier–Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objective function, as well as issues in the gradient computation via the adjoint equation method are discussed. Numerical results are presented for a model problem consisting of two counter‐rotating viscous vortices above an infinite wall which, due to the self‐induced velocity field, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wall‐normal velocity. Optimal controls for objective functions that target kinetic energy, heat transfer, and wall shear stress are presented along with the influence of control regularization for each case. Copyright © 2002 John Wiley & Sons, Ltd.     

RECONSTRUCTION OF BINARY RELATIONS FROM THEIR RESTRICTIONS OF CARDINALITY 2, 3, 4 and (n - 1) II

We shall prove here that any binary relation on a base E with cardinality n > 6 is reconstructible from its restrictions of cardinality 2, 3, 4 and (n - 1). This proof needs results of part I of this paper where we characterize any pair of relations R, R' which are 2-, 3- and 4-hypomorphic. As a corollary we obtain that any binary relation is (n - 4)-reconstructible (when n > 9).     

Chemical Reactivity of Tetrasulfur Tetranitride: Synthesis,Physical Properties,and Structural Characterization of the Amorphous Phase Cu7S4N4

Tetrasulfur tetranitride, S₄N₄, reacts with elemental Cu within inert solvents to a black‐blue material of approximate composition Cu₇S₄N₄ which is totally amorphous to X‐rays and which cannot be made crystalline by either thermal treatment or electron radiation. Cu₇S₄N₄ explodes if heated above 234 °C or when subjected to mechanical shock to eventually yield copper(I) sulfide; this together with the characteristic infrared spectrum of Cu₇S₄N₄ indicates the presence of molecular S₄N₄ units inside the amorphous phase. The metastable nature of Cu₇S₄N₄ is also mirrored by electron microscopy which furthermore allows the structural characterization of its degradation products. Based on experimental EXAFS data offering characteristic Cu—N and Cu—S distances, a theoretical crystalline approximant of Cu₇S₄N₄ was suggested and structurally optimized by density‐functional total‐energy calculations including periodic boundary conditions. This model incorporates a central S₄N₄ unit bonded to three shells of Cu atoms of different functionalities; in addition, a partial rupture of the S₄N₄ unit is likely to allow for a lowering of the total energy of the metastable phase. The latter observation supports the impossibility to make Cu₇S₄N₄ crystallize using ₄N₄ crystallize using whatever kind of measures.